NIMROD Simulations of a DIII-D Plasma Disruption

S. Kruger, D. Schnack (SAIC)

20th IAEA Fusion Energy Conference

November 2004

Vilamoura, Portugal

Outline

• Motivation– High-beta disruption discharge: #87009– Simple analytic theory

• NIMROD Modeling– Fixed boundary– Free-boundary using equilibria above marginal limit

Acknowledgements: D. Whyte, C. Sovinec, J. Callen (UW-M), A. Turnbull, M. Chu (GA)E. Held (USU)

DIII-D SHOT #87009 Observes a Plasma Disruption During Neutral Beam Heating At High Plasma Beta

Callen et.al, Phys. Plasmas 6, 2963 (1999)

Mode Passing Through Instability Point Has Faster-Than-Exponential Growth

• In experiment mode grows faster than exponential• Theory of ideal growth in response to slow heating

(Callen, Hegna, Rice, Strait, and Turnbull, Phys. Plasmas 6, 2963 (1999)):

(1 )c htβ β γ= +

2 2ˆ ( / 1)MHD cω γ β β=− −

Heat slowly through critical β:

Ideal MHD: ˆ( ) MHD ht tγ γ γ=

Perturbation growth:

( )d tdtξ γ ξ= 3/ 2 2 /3 2 /3 1/3

0 ˆexp[( / ) ], (3 / 2) MHD htξ ξ τ τ γ γ− −= =

DIII-D SHOT #87009 Observes a Mode on Hybrid Time Scale As Predicted By Analytic Theory

• Growth is slower than ideal, but faster than resistive

Callen et.al, Phys. Plasmas 6, 2963 (1999)

Turnbull, IAEA, Sorrento, Italy (2000)

Resistive MHD Equations Used to Numerically Model Disruption

• MHD Equations Solved:– Density Equation:

( ) VBJVVV 2 ∇−∇−×=⎟⎠⎞⎜

⎝⎛ ∇⋅+ μ∂∂ρ pt

– Resistive MHD Ohm’s Law:

{MHD

ResistiveMHD deal

JBVE η+×−= 321I

– Temperature Equations:

( ) ( )

TT

QTTtT

∇−−∇⋅−=

−+⋅∇−−=⋅∇+∇⋅+

⊥ )( :Currently

11

|||| κκκ

γγγ∂∂

α

ααααααα

bbq

qVV

– Momentum Equation

0 =⋅∇+ Vnτn∂∂

NIMROD’s Spatial Discretization Advantageous For Disruption Simulations

3≥

• Finite-Elements in Poloidal Plane, Pseudo-spectral in Toroidal Angle

• Can parallelize by FE blocks and by toroidal mode number

• Lagrangian elements of arbitrary polynomial degree (specified at runtime)– Spectral convergence needed for

realistic conditions:Error ~ hp+1

High S: Use polynomial degree ≥3

High ||perp: Use polynomial degree ≥4

See Sovinec et.al. JCP 195 (2004) 355

3≥

Initial Simulations Performed Using Fixed Boundary Equilibria

• Use q and pressure profile from experiment• Negative central shear

Fixed Boundary Simulations Require Going to Higher Beta

• Conducting wall raises ideal stability limit– Need to run near critical βN for ideal instability NIMROD gives

slightly larger ideal growth rate than GATO• NIMROD finds resistive interchange mode below ideal stability

boundary

0 100

5 10-2

1 10-1

1.5 10-1

2 10-1

3.5 4 4.5 5 5.5 6 6.5

γτ

A

βN

NIMROD

GATO

DCON marγinal poinτ

Resisτive Inτercηanγe Mode

Nonlinear Simulations Find Faster-Than-Exponential Growth As Predicted By Theory

• Impose heating source proportional to equilibrium pressure profile

∂P∂t

= ..... +γHPeq

€

⇒ βN = βNc 1+γH t( )

• Follow nonlinear evolution through heating, destabilization, and saturation

Log of magnetic energy in n = 1 mode vs. timeS = 106 Pr = 200 γH = 103 sec-1

Scaling With Heating Rate Gives Good Agreement With Theory

• NIMROD simulations also display super-exponential growth

• Simulation results with different heating rates are well fit by ξ exp[(t-t0)/τ] 3/2

• Time constant scales as

0

2

4

6

8

10

12

14

16

0 1 10

-6

2 10

-6

3 10

-6

4 10

-6

5 10

-6

6 10

-6

7 10

-6

(t - t

0

)

3/2

γ

H

=10

2

γ

H

=10

3

ln(W

m

W

m

0)

ξ eξp[(τ - τ

0

) τ]

32

, τ γ

MHD

-0.72

γ

H

-0.28

τ ~γMHD−0.72γH

−0.28

• Compare with theory:

τ =(3/2)2/ 3 ˆ γ MHD−2/ 3γh

−1/ 3

• Discrepancy possibly due to non-ideal effects

Log of magnetic energy vs. (t - t0)3/2

for 2 different heating rates

Free-Boundary Simulations Models “Halo” Plasma as Cold, Low Density Plasma

• Typical DIII-D Parameters:Tcore~10 keV Tsep~1-10 eVncore~5x1019 m-3 nsep~ 1018 m-3

• Spitzer resistivity: η~T-3/2

– Suppresses currents on open field lines

– Large gradients 3 dimensionally

• Requires accurate calculation of anisotropic thermal conduction to distinguish between open and closed field lines

Goal of Simulation is to Model Power Distribution On Limiter during Disruption

• Plasma-wall interactions are complex and beyond the scope of this simulation

• Boundary conditions are applied at the vacuum vessel, NOT the limiter.

– Vacuum vessel is conductor– Limiter is an insulator

• This is accurate for magnetic field:– Bn=constant at conducting wall– Bn can evolve at graphite limiter

• No boundary conditions are applied at limiter for velocity or temperatures.

– This allows fluxes of mass and heat through limiter

– Normal heat flux is computed at limiter boundary

Free-Boundary Simulations Based on EFIT Reconstruction

• Pressure raised 8.7% above best fit EFIT

• Above ideal MHD marginal stability limit

• Simulation includes:– n = 0, 1, 2– Anisotropic heat conduction

(with no T dependence)parperp=108

• Ideal modes grow with finite resistivity (S = 105)

First Macroscopic Feature is 2/1 Helical Temperature Perturbation Due to Magnetic Island

Magnetic Field Rapidly Goes Stochastic with Field Lines Filling Large Volume of Plasma

• Region near divertor goes stochastic first

• Islands interact and cause stochasticity

• Rapid loss of thermal energy results. Heat flux on divertor rises

Maximum Heat Flux in Calculation Shows Poloidal And Toroidal Localization

• Heat localized to divertor regions and outboard midplane

• Toroidal localization presents engineering challenges - divertors typically designed for steady-state symmetric heat fluxes

• Qualitatively agrees with many observed disruptions on DIII-D

Investigate Topology At Time of Maximum Heat Flux

• Regions of hottest heat flux are connected topologically

• Single field line passes through region of large perpendicular heat flux. Rapid equilibration carries it to divertor

• Complete topology complicated due to differences of open field lines and closed field lines

Initial Simulations Above Ideal Marginal Stability Point Look Promising

• Qualitative agreement with experiment: ~200 microsecond time scale, heat lost preferentially at divertor.

• Plasma current increases due to rapid reconnection events changing internal inductance

• Wall interactions are not a dominant force in obtaining qualitative agreement for these types of disruptions.

Future Directions

• Direct comparison of code against experimental diagnostics

• Increased accuracy of MHD model– Temperature-dependent thermal diffusivities– More aggressive parameters– Resistive wall B.C. and external circuit modeling

• Extension of fluid models– Two-fluid modeling– Electron heat flux using integral closures– Energetic particles

• Simulations of different devices to understand how magnetic configuration affects the wall power loading

Conclusions

NIMROD’s Advanced Computational Techniques Allows Simulations Never Before Possible

• Heating through β limit shows super-exponential growth, in agreement with experiment and theory in fixed boundary cases.

• Simulation of disruption event shows qualitative agreement with experiment.

• Loss of internal energy is due to rapid stochastization of the field, and not a violent shift of the plasma into the wall.

• Heat flux is localized poloidally and toroidally as plasma localizes the perpendicular heat flux, and the parallel heat flux transports it to the wall.